Dear Ali,

The Hinf synthesis is a method used to design optimal commands. It is essentially an optimization method, which takes into account a mathematical definition of the constraints with regard to the expected behavior in closed loop. The main advantage of the Hinf control is the ability to include in the same synthesis effort the concepts of classical control and robust control.  

The word "optimal" is used in its strictly mathematical sense because the synthesized command is that which will minimize the effect of the inputs / outputs of the system, which can be seen as "not optimal" by the operators (the optimization being relative to The desired objective).

The "infinity" in Hinf means that this type of control is designed to impose minimax restrictions in the sense of the decision theory (minimizing the maximum possible loss) in the frequency domain. The Hinfini standard of a dynamic system is the maximum amplification that the system can exert on the energy of the input signal. In the case of a MIMO system, this is equivalent to the maximum singular value of the system, which in the SISO case translates into the maximum value of the amplitude of its frequency response.

The theories of the synthesis H2 and the synthesis Hinf are well established. It is mainly the applications of this type of synthesis in the state space to increasingly complex practical cases that have developed in recent years. The confrontation of these very elegant but partially closed theories led to complementary observations. On the one hand, it is well known since the works presented, that the LQG regulator can present margins of robustness with respect to parametric, dramatically and arbitrarily weak uncertainties. On the other hand, the numerous recent applications of the Hinf  compensators to practical problems have also shown that it is relatively difficult to obtain with this type of corrector a satisfactory level of performance measured for example by the H2 standard. This is partly due to the fact that the Hinf synthesis is essentially based on the worst-case performance analysis whereas the H2 standard reflects an average performance. The idea of combining these two types of performance is therefore relatively natural, given that the H2 and Hinf approaches share the same formalism based on the standard problem.

Also i suggest attached files in topics for more details.

Best regards

Thank you for your answering. It was absolutely exhaustive. Do I understand right if the system is already stable by using H2 (LQG) it theoretically never be unstable? 

Theoretically this proposition is true, however this depends on the complexity of the system and its surroundings.